{
  "abstract": "Tikhonov regularization, kriging, and Gaussian-process regression share the same algorithm because, for a linear inverse problem with a Gaussian prior and Gaussian noise, the minimum-mean-square-error (MMSE) estimator is the solution of a regularized least-squares problem. We show that this equivalence survives the passage from vectors to generalized functions: the regularized least-squares estimators become generalized splines, and the Gaussian random vectors become generalized Gaussian processes (GGPs) on the dual of a nuclear space. A whitening operator $\\mathrm{L}: \\mathcal{S} \\to L_2$ specifies both the native Hilbert space $\\mathcal{H}_\\mathrm{L} \\subset \\mathcal{S}'$ of the splines and a family of GGPs, which solve a stochastic differential equation subject to boundary conditions. We prove that the Gaussian spaces of these processes are congruent to the core $\\mathrm{L}^\\ast(L_2)$ of the native space, and that their MMSE estimators from finitely many linear measurements are generalized splines. Our framework extends the classical correspondences of Kimeldorf-Wahba, Parzen and Kailath in two respects: our estimators form a broader class that includes generalized functions, and we give a proper Bayesian treatment of the case where the null space of $\\mathrm{L}$ is non-trivial. The framework gives a precise meaning to statements such as ``fractional splines are the optimal estimators of fractional Brownian motion,'' and it provides a rigorous Bayesian formulation of infinite-dimensional linear inverse problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06233",
  "author_total": 1,
  "authors": [
    {
      "name": "Michael Unser"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/02k89j8kg39qh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06233",
  "fulltext_url": null,
  "id": "02k89j8kg39qh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06233",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06233",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/02k89j8kg39qh.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06233",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06233"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Variational Splines and Generalized Gaussian Processes",
  "title_text": "Variational Splines and Generalized Gaussian Processes",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:32Z",
  "url": "https://research.draconitic.com/paper/02k89j8kg39qh",
  "venue": "arXiv",
  "versions": []
}